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Journal of Solid Mechanicsand MaterialsEngineering

Vol.5, No.11, 2011

Mechanism for a Bolt and Nut Self Looseningunder Repeated Bolt Axial Tensile Load

Tomotsugu SAKAI Toyota Industries Corporation, Technical Learning Center

3-217 Ebata-cho, Obu-shi,Aichi,474-0035 JapanE-mail: [email protected]

AbstractThe mechanism for a bolt and nut self loosening under repeated bolt axial tensile loadhas yet to be clarified, despite much investigation into this phenomena. In this paper,the self loosening mechanism is derived from basic strength of materials equations,which results in the following conclusions. (1) When load is applied, a slip occurson the screw thread surface, and the bolt shank twists clockwise, that is it descendson the lead angle of the screw thread. At the end of this process, counterclockwiserestitution torque TS 1 is generated by the twisted angle of the bolt shank. However,there is no rotation of the nut. (2) When the load is released, if TS 1 exceeds thefriction torque TW0 on the nut bearing surface generated by the decreased bolt axialtension, a slip occurs on the nut bearing surface, and the bolt and the mating nut rotatecounterclockwise as one unit. To satisfy the relationship TS 1 > TW0, the maximumbolt axial tension F1 must be larger than cF0. Here F0 is the minimum bolt axialtension and the coecient c depends on the bolt shape and the friction coecient.The rotational behavior of the bolt and the nut derived from this analysis concurs withexperimental and FEM calculation results of other researchers. For steel joints, it isbelieved that rotational loosening rarely occurs when there is no separation betweenjoined parts.

Key words : Fixing, Screw, Bolt, Nut, Loosening Mechanism, Tensile Load, Slip,Contact Surface, Friction

1. Introduction

Much research has been conducted into the self loosening mechanism of bolts and nutsfor various loads applied to the bolt axis. The main loads are the shear load, torsional loadand tensile load. The self loosening mechanism for a shear load joint has been investigatedby Yamamoto et al(1), Sakai(2), Kasei et al(3) and Pai et al(4), and for torsional load joint bySakai(5), and for tensile load joint by Goodier et al(6), Sato et al(7) and Izumi et al(8).

In summary of selected papers on the self loosening mechanism for a tensile load joint,which is investigated here more precisely, the following has been noted. First, Goodier et al(6)pointed out that a radial micro slip is generated both on the screw thread surface and on thebearing surface during the bolt axial tension change as the result of the radial contraction ofthe bolt and the radial expansion of the nut by the amount of Poissons ratio deformation, andderived an equation for loosening torque. But, the self loosening mechanism was not con-sidered. Sato et al(7) derived the equation to calculate the loosening angle of the bolt and nutbased on the micro slip locus on the screw thread surface and on the bearing surface assumingthat the descending micro slip occurs both when load is applied and released. Izumi et al(8)calculated the detailed loosening behavior of the bolt and nut under axial tension load usingthe three-dimensional finite element method, which provided very useful data for analysis ofthe self loosening mechanism, but did not clarify it in classical mechanics. The assumption bySato that a descending micro slip occurs both as load is applied and released does not concur

Received 15 July, 2011 (No. 11-0410)[DOI: 10.1299/jmmp.5.627]

Copyright c 2011 by JSME

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Journal of Solid Mechanicsand Materials Engineering

Vol.5, No.11, 2011

with Izumis calculation results.In this paper, based on the strength of materials, the twisting or rotational behavior of the

bolt and nut during axial loading and release are analyzed, and the self loosening mechanismis derived from basic strength of materials equations.

Nomenclature

F : Bolt axial tensionF0: Initial bolt axial tension=The minimum bolt axial tension in operationF1: The maximum bolt axial tension in operation: Flank angle : Lead angle of thread: Load factor P: Thread pitchd2: Pitch diameterdW : Equivalent friction diameter of the bearing surfaceL : Distance between the bearing surfaces of bolt and nutS : Friction coecient on the screw thread surfaceW : Friction coecient on the bearing surfaceTS: Descending clockwise torque that makes the bolt screw thread slip down

on the screw thread surface at F = F (Friction torque is subtracted.)TS 1: Descending clockwise torque of the bolt at F = F1(Friction torque is subtracted.)TS 0+: Residual clockwise torque generated on the bolt by the initial tightening to F = F0

(Friction torque is added.)TS T : Torque to make the bolt screw thread slip up on the screw thread surfaceTW : Friction torque on the bearing surface0+: Twist angle of the bolt shank generated by the initial tightening to F = F01: Twist angle of the bolt shank at the end of loading, F = F1F : Final residual angle of the bolt shankR: Counterclockwise returning angle of the bolt and the nut

during the unloading processL: Loosening angle of the nut per one cycle

2. Theoretical analysis of the mechanism for self looseningunder repeated bolt axial tensile load

2.1. Theoretical analysis modelA right-hand screw thread with a nut is shown in Fig.1. Under the condition that the bolt

axis is fixed, if the nut rotates clockwise the bolt axial tension increases (the bolt is tightened),and if the nut rotates counterclockwise the tension decreases (the bolt loosens). On the otherhand, under the condition that the nut is fixed, if the bolt axis rotates clockwise the bolt screwthread ridge goes down on the lead angle and the bolt loosens, whereas if the bolt rotates

Fig. 1 Nut with a right-hand threadRotational direction is the view from the nut side.Here, a triple-start thread with a large lead angle is shownto assist understanding.

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counterclockwise, the bolt tightens. Here, the rotational direction is defined by the view fromthe nut side. When a fluctuating tensile load is applied to parts joined by a bolt and nut, the boltaxial tension varies with the applied load. Under these circumstances, the rotational looseningof a bolt and nut, that is, the descending slip of the bolt screw thread on the nut screw threadis being investigated based upon basic strength of materials equations. For this investigation,the period during which the bolt axial tension is increased by increasing the applied tensileload, is called the loading process, and the period during which the bolt axial tension isdecreased, is called the unloading process.

2.2. Behavior during the loading processWhen bolt axial tension F is applied, a tangential force W = F tan is generated between

the bolt screw thread and the nut screw thread because the screw thread surface is inclinedtoward the plane perpendicular to the bolt axis by the lead angle , and the bolt screw threadand the nut screw thread push (jostle) each other [Fig.2]. For a right-hand thread, the bolt istwisted clockwise, while the nut is pushed back counterclockwise. Namely, this tangentialforce tends to twist both the bolt and nut in their loosening directions.

During the loading process, it is important to first determine the location where the slipoccurs, on the screw thread surface or on the bearing surface.

Fig. 2 Relationship between bolt axial tension F and circumferential force W

2.2.1. Theoretical study of slip surface(a) Condition for slippage to occur only on the screw thread surface and not on thebearing surface In order for the slip to occur on the screw thread surface, the tangentialforce F sin on the screw thread surface must be larger than the friction force S F cos / coson the screw thread surface. Here, S is the friction coecient on the screw thread surface, is a flank angle.

F sin > F cos cos

S (1) S < tan cos (2)

Using the relationship tan = P/d2, Eq.(2) can be also expressed as follows (P is the threadpitch, and d2 is the pitch diameter).

P d2

cosS > 0 (3)

According to Table 1 which shows the critical values S cr of S to satisfy Eq.(2), S mustbe less than at least 0.05 to satisfy Eq.(2) (Here, thread sizes range from M6 to M20. Thisis applied to the following.). Hereafter calculation results are shown only when Eq.(2) issatisfied, that is S 0.05.

Next, for the bolt screw thread to slip down on the screw surface, the torque TS mustbe smaller than the frictional resistant torque TW on the bearing surface (Otherwise, the slipwould occur on the bearing surface).

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Table 1 Critical friction coecient S cr and cr [Eq.(2) and Eq.(6)]

Dimensions of bolt and nut Critical friction coecient

Nominal thread For slippage to occur To satisfy self-supporting

designation dW on screw thread surface condition of screw threadS cr cr

M6 7.72 0.0515 0.0229

M8 10.28 0.0479 0.0214

M10 12.81 0.0458 0.0206

M12 15.03 0.0444 0.0202

M14 17.39 0.0434 0.0199

M16 19.84 0.0375 0.0173

M18 22.51 0.0421 0.0192

M20 24.96 0.0375 0.0172

dW : Equivalent friction diameter of bearing surface for a standard hexagon headwith washer face.(This applies to the following tables.)

TS =d22

(F sin F cos

cosS

) 1cos

< TW =dW2

FW (4)

W >1

dW

(P d2

cosS

)(5)

Here, dW is the equivalent friction diameter of the bearing surface, W is the friction coecienton the bearing surface. Eq.(5) is equal to Eq.(6), that is, the self-supporting condition of screwthread.

d2cos

S P+ dWW > 0 (6)

Therefore, if the self-supporting condition of screw thread is satisfied and the joined parts existas one piece, Eq.(5) is always satisfied and no slip will occur on the bearing surface beforea slip occurs on the screw thread surface. Assuming S = W ,the critical friction coecientcr that satisfies the self-supporting condition is calculated and shown in Table 1 (The bearingsurface is assumed to be a standard washer faced hexagon which is easily loosened.).(b) Condition for slippage to occur on the bearing surface Under this condition, theinequality sign of Eq.(5) is reversed and the self-supporting condition is not satisfied. There-fore, no slippage occurs on the bearing surface during the loading process in a normal boltedjoint.

From the above, it can be concluded that slippage occurs only on the screw thread surfaceduring the loading process in a normal joint.2.2.2. Deformation of bolt and nut during the loading process During the loadingprocess, a nut is compressed by the force applied to the screw thread ridges causing it tocontract axially and expand radially due to Poissons ratio eect. Similarly, Poissons ratioeect causes a bolt stretched by the increasing axial tension to elongate axially, and thencontract radially. As a result, relative micro slippage occurs radially both on the screw threadsurface and on the bearing surface(6)(9)(10).During the unloading process, as both deformationdirections are inversed, this radial micro slippage also occurs.

Therefore, if the bolt axial tension changes, the radial micro slippage is generated bothon the screw thread surface and on the bearing surface.2.2.3. Friction coefficient between surfaces which slide relative to each other(Concept of the friction circle) If a small force W is applied to a body being pushed byanother large force W and is sliding to the same direction as W, the friction coecient W

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in the direction of W becomes very small and approaches 0. The reason is as follows. Wis larger than the friction force when another small force W is applied (Here, for simplicity,W is perpendicular to W.). If the resultant force of W and W is greater than the frictionforce (a resultant force outside the friction circle. Fig.3(a)), slippage occurs in the directionof the resultant force. That means the small force W causes slippage in the direction of W .As the apparent friction coecient in the direction of W is calculated by W /F (F is thevertical force to the sliding surface), if W is very small, the apparent friction coecient in thedirection of W becomes very small. This is called the friction circle concept.

Fig. 3 Friction circle

W and W on the screw thread surface are shown in Fig.3(b). W is the radial forcegenerated by the radial micro slippage between the bolt and nut. W is the circumferentialforce generated by the descending clockwise torque TS. If the radial micro slippage betweenthe bolt and nut occurs, the friction coecient S in the circumferential direction on the screwthread surface becomes very small.2.2.4. Summary of behaviors during the loading process From the above, when the boltaxial tension changes, radial micro slippage is generated on the screw thread surface, and thefriction coecient S in the circumferential direction on the screw thread surface becomesvery small. As a result, Eq.(2) holds, and descending slippage occurs on the screw threadsurface during the loading process.2.2.5. Condition to produce clockwise torsion on the bolt When the bolt axial tensionincreases to F1, after loading, descending clockwise torque TS 1 is generated on the bolt.When torque TS 1 is greater than the residual clockwise torque TS 0+,generated by the initialtightening to F0, the bolt further twists clockwise.

TS 1 =F12

(P d2

cosS

)> TS 0+ =

F02

(P+

d2cos

S

)(7)

F1F0>

P+ d2

cosS

P d2

cosS= a (8)

Therefore, if F1 is larger than aF0 = FTcr, a clockwise twisted deformation is added the boltduring the loading process.2.2.6. Twist angle of bolt When a bolt is tightened to axial tension F0, the twist angle0+ of the bolt shank is expressed as follows.

0+ =32LTS 0+d4GB

=16LF0d4GB

(P+

d2cos

S

)(9)

Here, L is the distance between the bearing surfaces of the bolt and nut, d is the diameter of thebolt shank, and GB is the modulus of rigidity of the bolt material. When the bolt axial tensionis increased to F1 during the loading process, the clockwise torque of the bolt is TS 1.Thetwist angle 1 of the bolt at F1 can be expressed as follows.

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For F1 aF0 = FTcr (a > 1)

1 = 0+ (10)

For F1 aF0 = FTcr (a > 1)

1 =32LTS 1d4GB

=16LF1d4GB

(P d2

cosS

)(11)

Therefore, the clockwise twist angle of the bolt shank increases by when the bolt axialtension increases from F0 to F1.

= 1 0+ = 16Ld4GB

{P

(F1 F0) d2Scos

(F1 + F0)}

(12)

2.3. Behavior during the unloading processWhen the applied tensile force is decreased, the bolt axial tension also decreases to F.

This leads to the occurrence of two possibilities. (a) When the bolt shank is twisted clock-wise by 1, at the point where axial force F1 is applied, counterclockwise restitution torqueTS 1 is generated. On the other hand, TS T is required for the counterclockwise ascendingslippage on the screw thread surface to occur when the bolt axial tension decreases to F. IfTS 1 overcomes the torque TS T , the bolt slips against the screw thread surface and rotatescounterclockwise (tightening rotation). (b) If TS 1 overcomes the friction torque TW on thenut bearing surface at F, slippage occurs on the nut bearing surface and both the bolt andthe nut rotate as a unit counterclockwise (In this case, there is no slippage against the screwthread surface and therefore no relative rotation between the bolt and nut occurs.). It is un-known which of these two cases, (a) or (b), will occur. Therefore, the mechanism is furtherinvestigated and discussed.2.3.1. Investigation of slippage surface When the bolt shank is twisted clockwise by1 , through application of axial force F1 , the counterclockwise restitution torque TS 1 is asfollows.

TS 1 =F12

(P d2

cosS

)(13)

Torque TS T and Torque TW are as follows.

TS T =F2

(d2

cosS +

P

)(14)

TW =FWdW

2(15)

(a) Condition for ascending slippage to occur on the screw thread surface For boltaxial tension F ,when the resistance torques TS T and TW under counterclockwise rotation onthe screw thread surface and on the nut bearing surface are compared, TS T must be smallerthan TW (Otherwise slippage occurs on the bearing surface.).

W >1

dW

(d2

cosS +

P

)(16)

In Table 2, the minimum values of W/S for Eq.(16) to hold are shown. From Table 2,therelationship W > 1.6S is required for the equation to hold. Assuming that S and W are in-dependent and follow a normal distribution, and the mean values for both friction coecientsare equal, even when S and W scatter very widely (the variation coecients of S and Ware 0.10), the probability of Eq.(16) to hold resulting in an ascending slippage is 0.13% orless. This means that the ascending slippage on the screw thread surface, that is, tighteningrotation of the bolt, rarely occurs during the unloading process for normal cases in which Sand W are nearly equal.

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Table 2 Critical friction coecient ratio W/S for slippage to occuron screw thread surface during the unloading process [Eq.(16)]

Nominal Critical friction coecient ratio

thread W/Sdesigna- dW S = 0.05 S = 0.04 S = 0.03 S = 0.02

tion

M6 7.72 1.63 1.83 2.18 2.86

M8 10.28 1.58 1.78 2.10 2.74

M10 12.81 1.56 1.75 2.06 2.68

M12 15.03 1.58 1.76 2.07 2.69

M14 17.39 1.58 1.76 2.06 2.67

M16 19.84 1.50 1.66 1.93 2.46

M18 22.51 1.55 1.72 2.02 2.61

M20 24.96 1.49 1.65 1.91 2.45

In the shaded areas,either Eq.(2) or Eq.(6) is not satisfied.(This applies to the following tables as well.)

When the mean value of W is much greater than that of S and Eq.(16) holds, to permitthe tightening rotation of bolt, the next equation must hold.

TS 1 =F12

(P d2

cosS

)> TS T =

F2

(P+

d2cos

S

)(17)

FF1

WdW(P d2

cosS

) = c = 1b (26)

2.4.3. Loosening angle L per one cycle under constant load change When the be-havior shown in Fig.5(a) is repeated, the loosening angle L per one cycle is derived bysubtracting the residual angle F(F = F0) from the angle 1 at F = F1.

L = 1 F(F = F0) = 16Ld4GB

{F1

(P d2

cosS

) F0WdW

}(27)

3. Comparison of test and FEM results

3.1. Behavior during the loosening processThe behavior that the relative loosening rotation between the bolt and nut is generated

during the loading process, and that the relative loosening rotation does not occur duringthe unloading process as shown in Fig.5 (b) concurs with Tsumuras experimental results(11).Furthermore, the rotational behavior of the bolt and nut shown in Fig.5 (a) concurs with theFEM calculation results of the bolt and nut during the loading and unloading process presentedby Izumi et al(8).

3.2. Loosening angleThis analysis cannot determine the friction coecient of fasteners during the actual loos-

ening process, though it is assumed that this will be a very small value. Therefore, the frictioncoecient resulting in the same loosening angle as that in Satos experiments(7) is calculatedbackward by Eq.(27), and the validity of this analysis is checked against the calculated frictioncoecient value.

The friction coecient that provides the same loosening angle as Satos experimentalresults, 0.0035/cycle, is 0.033 as derived from Eq.(27). This analysis is judged to be appro-priate, because this value of 0.033 satisfies both requirements from Eq.(2) (S < 0.038) andEq.(6) (S = W > 0.017).

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3.3. Value of F1/F0 for rotaitional loosening to occurIn both Satos experiment(7) and Izumis calculation(8), it is observed that rotational loos-

ening occurs under the conditions: M20, F1 = 29.4 kN and F0 < 12.4 kN. This means thevalue of c = F1/F0 is greater than 2.37. The friction coecient which satisfies F1/F0 > 2.37is more than 0.025 as understood from Eq.(26).

The above-mentioned friction coecient of 0.033 satisfies this condition S = W >0.025, and falls into the range that fulfills the following three conditions: S < 0.038, S =W > 0.017 and S = W > 0.025.

From the above, it can be concluded that there are no inconsistencies in this analysis.

4. Reason why rotational loosening rarely occurs in actual use

In actual use, rotational loosening rarely occurs(12)(13). The reason for this is consideredin the following.

When bolt axial tensile load is applied to joints, if there is any separation between thejoined elements, the bolt load sharing increases and fatigue failure of the bolt easily occurs.Therefore, it is a basic design feature to prevent any separation between joined elements. Thecondition that joined elements are not separated is expressed as follows. Here, is the loadfactor,W is a tensile load applied to the joint and c = F1/F0.

F1 F0 = W, F0 > (1 )W

c 4.0. Therefore, based on this thesis, it can be con-cluded that rotational loosening does not occur if the joined parts do not become separated,including the aluminum alloy structures, for which is generally less than 0.7.

5. Conclusions

The mechanism for self loosening of a bolt and nut under repeated axial tensile load wasanalyzed based on strength of materials equations, and the following conclusions were arrivedat.

( 1 ) During the loading process (bolt axial tension F0 F1)As a result of radial micro slippage both on the screw thread surface and on the bearingsurface, S and W become very small values in accordance with the friction circle concept.The descending force component of F on the screw thread surface then exceeds the frictionforce and the bolt is twisted clockwise resulting in the twist angle of the bolt reaching to 1at F = F1. Under the self-supporting condition, slippage does not occur on the nut bearingsurface nor does the nut rotate.

( 2 ) During the unloading process (F1 F0)When the counterclockwise restitution torque TS 1, which is generated in the bolt axis bythe clockwise twist angle 1 during the loading process, exceeds the bearing surface frictiontorque TW , as F is decreased below F1/c, slippage occurs on the nut bearing surface and both

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the bolt and the nut rotate as a unit counterclockwise by R [Eq.(22)]. In this circumstance,there is no relative rotation between the bolt and nut.

( 3 ) Under constant change in axial tension (F0 F1)If F1 is larger than cF0, the counterclockwise loosening angle L [Eq.(27)] is generated onthe nut cycle by cycle which is then accumulated. The rotational loosening condition is suchthat the maximum bolt axial tension F1 must be greater than cF0 , here F0 is the minimumbolt axial tension and c is given by Eq.(26).

( 4 ) If S and W are approximately 0.03, the results of this analysis concur with Satosexperiments. And the rotational behaviors of the bolt and nut during the loosening processconcur with Izumis FEM calculation results. From the above mentioned fact, it is concludedthat the rotational loosening mechanism under repeated bolt axial tensile load has been clari-fied by this analysis.

( 5 ) For steel joints, it is believed that rotational loosening rarely occurs if there is noseparation between the joined parts.References

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( 2 ) Sakai T., Investigations of Bolt Loosening Mechanisms-1st Report, On the Bolts ofTransversely Loaded Joints-, Bulletin of the Japan Society of Mechanical Engineers,Vol.21, No.159(1978), pp.1385-1390.

( 3 ) Kasei S., Ishimura M. and Ohashi H., On Self-loosening of Threaded Joints in theCase of Absence of Macroscopic Bearing-surface Sliding-Loosening Mechanism un-der Transversely Repeated Force, Bulletin of Japan Society of Precision Engineering,Vol.54, No.7(1988), pp.1381-1386

( 4 ) Pai N.G. and Hess D. P., Three-dimensional finite element analysis of threaded fastenerloosening due to dynamic shear load, Engineering Failure Analysis, 9(2002), pp.383-402.

( 5 ) Sakai T., Investigations of Bolt Loosening Mechanisms-2nd Report, On the CenterBolts of Twisted Joints-, Bulletin of the Japan Society of Mechanical Engineers, Vol.21,No.159(1978), pp.1391-1394.

( 6 ) Goodier J.N. and Sweeney R.J., Loosening by Vibration of Threaded Fastenings, Me-chanical Engineering, Vol.67, No.12(1945), pp.798-802

( 7 ) Sato S., Hosokawa S. and Yamamoto A., Studies on Loosening Mechanism of BoltNut Units (2nd Report)-A Solution for Self-loosening Mechanism in the Repeated Ten-sile Loads-, Bulletin of Japan Society of Precision Engineering, Vol.51, No.8(1985),pp.1540-1546.

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( 9 ) Hosokawa S., Sato S. and Tsumura T., Deformation of Nut in Threaded Connection,Bulletin of Japan Society of Precision Engineering, Vol.51, No.10(1985), pp.1909-1913.

(10) Hosokawa S., Sato S., Miyata C. and Tsumura T., Contraction Deformation of BoltShank in Threaded Connection, Bulletin of Japan Society of Precision Engineering,Vol.53, No.11(1987), pp1726-1732.

(11) Tsumura T. and Sato S., Studies on Loosening Mechanism of Bolt Nut Unit (3rd Report) A consideration to the eect of screw thread shape on loosening under bolt axialloading (In Japanese), Proceedings of the 1978 Spring Annual Meeting of JapanSociety of Precision Engineering, (1978), pp.7-8.

(12) Sakai T., Investigations of Bolt Loosening Mechanisms (3rd Report)-On the Bolts Tight-

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ened over Their Yield Point-, Bulletin of the Japan Society of Mechanical Engineers,Vol.22, No.165 (1979-3), pp.412-419.

(13) Sakai T., Investigations of Loosening Characteristic of Connecting-Rod Cap Bolt (InJapanese), Transactions of the Japan Society of Mechanical Engineers, Vol.43, No.368(1977-4), pp.1454-1461.

(14) Sato S., Hosokawa S., Tsumura T. and Yamamoto A., On the loosening of bolt andnut under fluctuation in axial load (Evaluation of anti-loosening devices of every kindand its method) (In Japanese), Study Reports of Faculty of Engineering in KanagawaUniversity, Vol.19 (1981-3), P.31.

(15) Sauer J.A.,Lemmon D.C. and Lynn E.K.,Bolts-How to prevent their loosining-, MachineDesign, (1950-8), pp.133-139.

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